An improved lattice Boltzmann method with a novel conservative boundary scheme for viscoelastic fluid flows

TL;DR

This paper introduces an enhanced lattice Boltzmann method with a novel boundary scheme for simulating high Weissenberg number viscoelastic flows, improving stability and accuracy in complex geometries.

Contribution

It develops a new lattice Boltzmann framework with a conservative boundary scheme and improved constitutive equation handling for high Weissenberg number flows.

Findings

Successfully simulates flows at Wi up to 10,000.

Demonstrates improved accuracy and stability over existing methods.

Effectively handles complex boundary conditions in viscoelastic flows.

Abstract

The high Weissenberg number problem has been a persistent challenge in the numerical simulation of viscoelastic fluid flows. This paper presents an improved lattice Boltzmann method for solving viscoelastic flow problems at high Weissenberg numbers. The proposed approach employs two independent two-relaxation-time regularized lattice Boltzmann models to solve the hydrodynamic field and conformation tensor field of viscoelastic fluid flows, respectively. The viscoelastic stress computed from the conformation tensor is directly embedded into the hydrodynamic field using a newly proposed local velocity discretization scheme, thereby avoiding spatial gradient calculations. The constitutive equations are treated as convection-diffusion equations and solved using an improved convection-diffusion model specifically designed for this purpose, incorporating a novel auxiliary source term that eliminates the need for spatial and temporal derivative computations. Additionally, a conservative non-equilibrium bounce-back (CNEBB) scheme is proposed for implementing solid wall boundary conditions in the constitutive equations. The robustness of the present algorithm is validated through a series of benchmark problems. The simplified four-roll mill problem demonstrates that the method effectively improves numerical accuracy and stability in bulk regions containing stress singularities. The Poiseuille flow problem validates the accuracy of the current algorithm with the CNEBB boundary scheme at extremely high Weissenberg numbers (tested up to Wi = 10,000). The flow past a circular cylinder problem confirms the superior stability and applicability of the algorithm for complex curved boundary problems compared to other existing common schemes.